Genus bounds for knot polynomials of Lie superalgebras

Abstract

Knot polynomials colored by typical representations of Lie superalgebras of type I (except $\mathfrak{psl}(n|n)$) have two variables $q$ and $t$, the latter corresponding to the complex-valued weight of the distinguished odd root. We prove that for every typical representation of a Lie superalgebra of type I, the $t$-degree of the knot polynomial is at most the number of odd roots times the genus of the knot. A complimentary bound being at least the number of odd roots times degree of the Alexander polynomial can be obtained from a specialization at $q=1$. These two bounds become equalities when the Alexander polynomial detects the genus of the knot, as is the case for alternating knots and fibered knots.